Laser-speckle-visibility acoustic spectroscopy in soft turbid media

We image the evolution in space and time of an acoustic wave propagating along the surface of turbid soft matter by shining coherent light on the sample. The wave locally modulates the speckle interference pattern of the backscattered light, which is recorded using a camera. We show both experimentally and theoretically how the temporal and spatial correlations in this pattern can be analyzed to obtain the acoustic wavelength and attenuation length. The technique is validated using shear waves propagating in aqueous foam. It may be applied to other kinds of acoustic waves in different forms of turbid soft matter such as biological tissues, pastes, or concentrated emulsions.

Figure 1

Experimental setup. An expanded laser beam shines on the top surface of a turbid foam sample in the region highlighted in red. The volume where the foam is probed by the light extends a few scattering mean free paths into the sample volume. The backscattered light forms a speckle interference pattern that is recorded using a camera. It is equipped with an objective, allowing speckle fluctuations to be detected as a function of the position at the sample surface. The acoustic emitter consists of a plate inserted into the sample and translated sinusoidally along the $x_2$ direction, with frequency $f$ and displacement amplitude $A_0$. This plate generates an acoustic shear bulk wave that propagates through the scattering volume in the $x_1$ direction.

Figure 2

Typical foam structure viewed at the surface of the sample. The average bubble diameter is $d=75\mu$m and the gas volume fraction is $\phi =94.2\%$.

Figure 3

Speckle visibility $V(T,t,x_1)$ and shear strain ${\varepsilon }_{12}$ versus the phase of the acoustic wave $\kappa x_1-\omega t$. (a) The continuous curves are predicted by Eq. (21) for different strain amplitudes ${\varepsilon }_0=n\times {10}^{-3}$, where $n$ takes integer values ranging from 1 (top curve) to 10 (bottom curve) and an exposure time such that $\omega T=\pi /10$. The visibility decreases monotonically as a function of $n$. (b) The continuous curves are predicted by Eq. (21) for a strain amplitude ${\varepsilon }_0={10}^{-3}$ and different exposure times such that $\omega T=n\pi /10$, where $n$ takes integer values ranging from 1 (top curve) to 10 (bottom curve). The visibility decreases monotonically with $n$. In both (a) and (b) the red dotted line indicates the prediction of the linearized expression (22) for ${\varepsilon }_0={10}^{-3}$ and $\omega T=\pi /10$. In all cases, we assume $\gamma =1.5$, $\beta =1$, ${\ell }^{*}=100\mu$m, and a light wavelength of 500 nm. (c) The oscillating shear strain is illustrated for an amplitude ${\varepsilon }_0={10}^{-3}$.

Figure 4

Space-time plot of the visibility as a function of time $t$ and propagation distance $x_1$, derived from snapshots at successive instants. The gray level is proportional to the level of visibility as indicated by the grayscale chart. The bright stripes correspond to instants and positions where the speckle visibility is maximal [see Eq. (22)], i.e., where the acoustic wave signal goes through an extremum (see Fig. 3). The acoustic wave frequency is $99.5$ Hz, the amplitude displacement $A_0=9.2\mu$m, and the exposure time $T=5.0$ ms. The stroboscopic acquisition parameters are $R=100$ frames per second, $N=1$ and, $1/\delta N=-200$ [see the text for the definition of these notations and Eq. (23)]. The sample is Gillette foam, with average bubble diameter $d=48\mu$m and gas volume fraction $\phi =(93.7\pm 0.4)\%$.

Figure 5

Phase ${\Phi }_V$ of the visibility Fourier component at angular frequency $\omega$ calculated from the space-time plot shown in Fig. 4 as a function of propagation distance. The data are represented by the symbols. The line corresponds to a linear regression to the data: The slope is (0.358 $\pm$ 0.002) rad mm${}^{-1}$. This is equal to twice the acoustic wave number $\kappa =0.18$ rad mm${}^{-1}$. We deduce the acoustic phase velocity $c=\omega /\kappa =3.5$ m s${}^{-1}$.

Figure 6

Variation of the function $S$ defined in Eq. (26) with the parameter $\omega T$.

Figure 7

(a) Visibility $\overline{V}({\varepsilon }_0,\omega T)$, defined in Eq. (25), with $\omega T=4\pi$, versus propagation distance $x_1$, measured with a foam sample ($d=51\mu$m, $\phi =$ 93.8$\%$) at a frequency $f=100$ Hz and exposure time $T=20$ ms. The symbols correspond to different displacement amplitudes as labeled in the graph. The speckle data are binned using 28 pixels per bin. The error bars are smaller or of the size of the symbols. The two straight lines are exponential fits, yielding attenuation lengths $l_A=10.8$ mm for $A_0=4.5\mu$m and $l_A=12$ mm for $A_0=3.2\mu$m. (b) The same data collapsed on a master curve after an offset $l_{A}ln\left(\xi \right)$ is applied to the distance $x_1$, with $l_A=11.9$ mm. The reference curve corresponds to the amplitude $A_0=9.2\mu \text{m}$. The symbols are the same as in (a).

Figure 8

Acoustic wavelength versus frequency measured for aqueous foam (Gillette shaving foam) with different average bubble diameters $d$ as labeled. The open symbols correspond to the data measured using laser speckle visibility spectroscopy (this study) while the closed symbols are deduced using Eqs. (10) and (11) from previous rheological measurements performed for the same foam and bubble sizes [21]. The solid lines represent the wavelength predicted by the viscoelastic constitutive model (28) with the following parameters: $d=45\mu$m, $G_0=222$ Pa, $f_c=26$ Hz, and ${\eta }_0=0.26$ Pa s (blue); $d=62\mu$m, $G_0=160$ Pa, $f_c=12$ Hz, and ${\eta }_0=0.21$ Pa s (red); and $d=75\mu$m, $G_0=136$ Pa, $f_c=8$ Hz, and ${\eta }_0=0.18$ Pa s (green). The inset shows the phase velocity $c=\lambda f$ for the same data and model predictions, with the same legend for the symbols.

Figure 9

Attenuation length versus frequency measured for the same foam and bubble diameter (as labeled) as in Fig. 8. The open symbols correspond to laser-speckle-visibility spectroscopy data (this study) while the closed symbols are deduced using Eqs. (10) and (11) from previous rheological measurements performed with a rheometer for the same foam and bubble sizes (data from Ref. [21]). The solid lines represent the attenuation length predicted by the viscoelastic model (28) with the same parameters as in Fig. 8. The inset shows the frequency versus wave number for the same samples, with the same legend for the symbols.